L(s) = 1 | + 0.759·3-s + 3.44·5-s + 4.51·7-s − 2.42·9-s + 1.79·11-s + 0.270·13-s + 2.61·15-s + 2.22·17-s + 4.30·19-s + 3.42·21-s − 2.66·23-s + 6.86·25-s − 4.12·27-s + 0.950·29-s − 1.11·31-s + 1.36·33-s + 15.5·35-s + 11.3·37-s + 0.205·39-s − 5.32·41-s − 1.81·43-s − 8.34·45-s − 5.51·47-s + 13.3·49-s + 1.68·51-s − 11.7·53-s + 6.17·55-s + ⋯ |
L(s) = 1 | + 0.438·3-s + 1.54·5-s + 1.70·7-s − 0.807·9-s + 0.540·11-s + 0.0750·13-s + 0.675·15-s + 0.539·17-s + 0.987·19-s + 0.748·21-s − 0.555·23-s + 1.37·25-s − 0.793·27-s + 0.176·29-s − 0.200·31-s + 0.237·33-s + 2.62·35-s + 1.86·37-s + 0.0329·39-s − 0.831·41-s − 0.276·43-s − 1.24·45-s − 0.804·47-s + 1.91·49-s + 0.236·51-s − 1.61·53-s + 0.833·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.178928604\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.178928604\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
good | 3 | \( 1 - 0.759T + 3T^{2} \) |
| 5 | \( 1 - 3.44T + 5T^{2} \) |
| 7 | \( 1 - 4.51T + 7T^{2} \) |
| 11 | \( 1 - 1.79T + 11T^{2} \) |
| 13 | \( 1 - 0.270T + 13T^{2} \) |
| 17 | \( 1 - 2.22T + 17T^{2} \) |
| 19 | \( 1 - 4.30T + 19T^{2} \) |
| 23 | \( 1 + 2.66T + 23T^{2} \) |
| 29 | \( 1 - 0.950T + 29T^{2} \) |
| 31 | \( 1 + 1.11T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 + 5.32T + 41T^{2} \) |
| 43 | \( 1 + 1.81T + 43T^{2} \) |
| 47 | \( 1 + 5.51T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 8.37T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 4.49T + 71T^{2} \) |
| 73 | \( 1 + 7.98T + 73T^{2} \) |
| 79 | \( 1 - 7.10T + 79T^{2} \) |
| 83 | \( 1 - 0.878T + 83T^{2} \) |
| 89 | \( 1 + 3.79T + 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.048667437700339759515723504673, −7.65233824106873340312679020575, −6.48813546292856949304312477032, −5.87531726113929361162436878079, −5.24406991178043717875693495769, −4.67176336781458028453710621103, −3.51490252856280071360283618995, −2.59167341895830904674508093696, −1.82757663894017101704883966146, −1.18890497266605812563040185310,
1.18890497266605812563040185310, 1.82757663894017101704883966146, 2.59167341895830904674508093696, 3.51490252856280071360283618995, 4.67176336781458028453710621103, 5.24406991178043717875693495769, 5.87531726113929361162436878079, 6.48813546292856949304312477032, 7.65233824106873340312679020575, 8.048667437700339759515723504673